From Strings to Black Holes

From Strings to Black Holes

Lorenzo Pieri

"Stringy Origin of 4d Black Hole Microstates", [1603.05169] M. Bianchi, J.F. Morales, L. Pieri

"Fuzzballs in general relativity: a missed opportunity", [1611.05276] L. Pieri

"More on microstate geometries of 4d black holes", [1701.05520] M. Bianchi, J.F. Morales,L. Pieri, N. Zinnato

Work in progress, S. Giusto, J.F. Morales, L. Pieri, R. Russo

Plan of the Talk

Classical Black Holes and the Information Paradox Stringy Origin of 4d Black Holes Microstates: From Open Strings on D-Branes to Supergravity Fields with scattering amplitudes

Smooth Horizonless geometries in Supergravity

Part I

Information Paradox

A pure state enters into a BH. The emitted radiation is thermal (no information), but is entangled with the BH. The emitted particle pairs do not depend on the state of earlier produced pairs (why? See later!). When the BH completely evaporates there is nothing to be entangled with. At the end, we have only

Information Paradox

The Black Hole Information Paradox arises since the particle pairs are created from the vacuum!

Information Paradox: Possible Resolutions

The paradox cannot be solved with small corrections to the semi-classical computation(Mathur), and it’s too late to extract the

information at the last stages of evaporation (Page Time). I Loss of unitarityHawking, Unruh, Wald

I Remnants, Baby UniverseSusskind

I Non Local interactions BH-radiationMaldacena-Susskind, Raju-Papadodimas

I _{Hairs in the asymptotic structure of spacetime}Hawking, Perry, Strominger

I _{The horizon is no more in an "information free vacuum"}

Fuzzball, Firewall

We will explore the last possibility. Rather than only solving an ad hoc problem, this resolution seems to emerge naturally from

BHs in String Theory: The Naive D1-D5

Black Holes in string theory can be constructed as bound states of intersecting (Dp-M)branes

Brane t x1 x2 x3 x4 y5 y6 y7 y8 y9

D1 − . . . . − . . . .

D5 − . . . . − − − − −

ds2= (H1H5)−1/2(−dt2+dy52) + (H1H5)1/2(dx21+. . .dx24) + H₁1/2H₅−1/2(dy2₆+. . .dy2₉)

F01m=∂mH₁−1 F0...5m=∂mH₅−1 eφ=H₁1/2H₅−1/2

Hn=1+ Qn

D1-D5 Fuzzball

ds2₌q H

1+K[−(dt−Aidx i₎2_+(dy

5+Bidxi)2)]+

q

1+L

H dx

2

i+

p

H(1+K)dy2 a

H−1=1+Q1

LT RLT

0 |~x−~dv_F(v)|2 K=

Q1

LT RLT

0

dv( ˙F(v))2 |~x−~_F(v)|2

Ai =−Q1

LT RLT

0

dvFi˙(v)

|~x−~_F(v)|2 dB=−?4dA v=t−y5

For instanceF1 =cos(ωv), F2 =sin(ωv), F3=F4=0.The metric has only a coordinate singularity on the curveF(v), indeed near curve we recover the metric of a Kaluza-Klein

monopole times regular spaces.

The D1D5 fuzzball is horizonless and regular. Instead of going past the "would-be horizon", the throat ends in smooth "cap"

depending onF(v), the hair which discriminate between different microstates. There is no "interior space".

Part II

Stringy Origin of 4d Black Holes Microstates: From Open Strings on D-Branes to Supergravity Fields with scattering

Stringy Origin of 4d Black Holes Microstates

Our goal is to recover the microstate geometries in supergravity from the underlying fundamental string theory description. In

particular we consider a bound state of 4 perpendicularly intersecting D3 branes onT6inIIB. (dual to D1-D5-KK-P ,

D23_{-D6 inIIAor M2-M5-P-KK inM-theory)} Brane t x1 x2 x3 y1 y˜1 y2 ˜y2 y3 ˜y3

D30 − . . . − . − . − .

D31 − . . . − . . − . −

D32 − . . . . − − . . −

D33 − . . . . − . − − .

We will indeed derive a 1:1 relation between open string condensates and fields in the bulk for a large class of 4d BPS

Mixed Open-Closed Scattering Amplitudes

The microstate geometries will be derived from mixed open-closed disk amplitudes, computing the emission rate of massless closed strings from open string condensates binding

different stacks of branes.

Closed String Fields

gMN,bMN,C(MNPQ4)

Open String Fields

From Amplitudes to Supergravity Fields

A(k)∝R _d2+n_z

VCKVhWclosed(z,¯z)Vopen(z1). . .Vopen(zn)i We will work at the leading order ings, take all open string momenta equal to zero and the closed string momentumkonly in non compact space directions. This limit is well defined only if the disk diagram cannot factorize via the exchange of open string states, so one must choose the polarization of the open strings in such a way that no factorization diagram is allowed.

The deviation from flat space of a closed field is extracted from the string amplitude:δΦ(˜ k) =−i

k2

_δA(k)

δΦ

δφ˜(k) =− i

k2

δA(h,k)

δh → δφ(x) =

Z _d3_k

(2π)2φ˜(k)e ikx

10d Supergravity Solution

ds2=−e2U(dt+w)2+e−2U P3

i=1dxi2+ P3

I=1

dy2

I

Ve2U_ZI +Ve2UZI˜e2I

C4=α0∧˜e1∧˜e2∧˜e3+β0∧dy1∧dy2∧dy3+ 1

2IJK (αI∧dyI∧˜eJ∧e˜K+βI∧˜eI∧dyJ∧dyK)

PI= KI

V, ZI =LI+

|IJK| 2

KJKK

V , µ= M

2 + LIKI

2V +

|IJK| 6

KIKJKK V2

e−4U _=Z

1Z2Z3V−µ2V2, bI =PI−_ZIµ, ˜eI =d˜yI−bIdyI

∗₃dA=dV ∗₃dwI =−d(KI) ∗3dv0 =dM

∗₃dvI =dLI ∗3dw=Vdµ−µdV−VZIdPI Ha ={V,LI,KI,M} I =1,2,3

RMN= 1

4·4!FMP1P2P3P4F

P1P2P3P4

Harmonic Multipole Expansion

LI ≈1+αD3 N_|x|I V≈1+αD3N_|x0_|

KI ≈αD3cKiI x

i

|x|3 M≈αD3cMi x

i

|x|3 αD3 =4πgs(α0)2/VT3

Ha(x) =ha+P∞n=0cai1...inPi1...in(x)

1

|x+a| =

P∞

n=0ai1. . .ainPi1...in(x)

P(x) = _|1_x| Pi(x) =−_|xxi_|3 Pij(x) =

3xixj−δij|x|2

|x|5

Pi1...in(x) =

R _d3_k

(2π)3eikxP˜i1...in(k) P˜i1...in(k) =

4π(i)n

n!k2 ki1. . .kin

ThePi1...in(x)are singular, but for an appropriate choice of the coefficientsci1...in the infinite sum of terms produce a fuzzy and

L Solution

V=L(x) M=KI =0 LI =1

The L solutions are geometries that fall-off at infinity asQi/r, corresponding to a single stack of branes. The "superposition" of this

solution together with K and M solutions will give the complete microstate of the SUGRA solution.

At linear order inαD3one finds:

δgMNdxMdxN = δ₂L h

dt2₋P3

i=1(dy2i −dx2i −d˜yi2) i

+. . . δC4=−δL∧dt∧dy1∧dy2∧dy3+A∧d˜y1∧dy˜2∧d˜y3+. . .

withδL=L−1 andAboth of orderαD3. One can take:

L=1+αD3N0

One boundary Amplitude

Vξ(φ)(xa) =P∞n=0ξi1...in∂Xi1(x1)

Qn a=2

R∞

−∞ dxa2π ∂Xia(xa)

WNSNS(z,¯z) =cNS(ER)MNe−ϕψMeikX(z)e−ϕψNeikRX(¯z)

ξ(φ) =P∞

n=0ξi1...inφ

i1_{. . . φ}in _E=h+b

A_NS−NS,ξ(φ)=hc(z)c(¯z)c(z1)iWNS−NS(z,¯z)Vξ(φ)

=i cNS tr(ER)ξ(k)

The asymptotic deviation from the flat metric can be extracted:

δ˜gMN(k) =

− i

k2

P∞

n=0

δA_NS−NS,φn δhMN =cNS

ξ(k)

k2 (ηR)MN After Fourier transform one finds agreement with SUGRA:

δgMN =R d

3_k

(2π)3δ˜gMN =−1₂(ηR)MNδL(x) In particular, for a single D3 brane at positionx=a:

K Solution

K3=−M=K(x) µ=0 LI =V=1 K1=K2=0 The K solutions are geometries that fall-off at infinity asQiQj/r2. They are associated to fermionic bilinears localized at the intersection

of two branes and in general they carry angular momentum. The open string condensates discriminate between different microstates.

At linear order inαD3one finds (∗3dw=−dK):

δgMNdxMdxN =−2dt w−2K dy3˜dy3+. . .

δC4= (K dt∧dy3−w∧d˜y3)∧(dy1∧d˜y2+d˜y1∧dy2) For example one can takeKto be

K≈ vixi_|

Two boundary Amplitude

ANS_µ2−NS =R

dz4hc(z1)c(z2)c(z3)i hV¯µ(z1)Vµ(z2)W(z3,z4)i Vµ¯(z1) = ¯µAe−ϕ/2SAσ2σ3 Vµ(z2) =µBe−ϕ/2SBσ2σ3

D

trµ¯(AµB)E= cO

3! vMNP(ΓMNP)AB vMNP∈10of SO(6)

ANS_µ2−NS= ₃1_!(ER)_MNkPvMNP

One can turn onvy3˜y33=−v12t=4πv, finding SUGRA fields δg2t =−v_|xx1_|3 δg1t=v_|x_x2_|3 δgy3˜y3 =−v

x3

M Solution

K2=M=M(x) µ=M LI =V=1 K1=K3=0 The M solutions are geometries that fall-off at infinity as

Q1Q2Q3Q4/r3. This factor appears in the entropy, i.e. the square root of theE7(7)quartic invariant ofN =8 SUGRA ind=4.

δgMNdxMdxN =2M

dy1˜dy1+dy3˜dy3

+. . .

δC4=−M dt∧ (dy1∧d˜y2∧dy3+d˜y1∧d˜y2∧dy˜3) +w2∧(dy1∧ dy2∧dy3+d˜y1∧dy2∧d˜y3) +. . .

The M solution are actually associated to the vanishing combinationcM_i +P3

i=1cKi =0, so they start as≈ r13 (no dipole

modes). In particular we take the harmonicMto be of the form: M ≈vij3xixj_|−_x|5δij|x|2

Four Boundary Amplitude

Consider the insertions of four fermionsµastarting on a D3-brane of type(a)and ending on a D3-branes of type(a+1)witha=0,1,2,3 (mod 4), in a cyclic order. The condensate is complex. Indeed, even if each intersection preservesN =2 SUSY (1/4 BPS), so that each fermionµacomes together with its charge conjugateµ¯a, the overall configuration preserves onlyN =1 SUSY (1/8 BPS), so that we have two pairs of opposite chirality.

ANS−NS

µ4 =hc(z1)c(z2)c(z4)i R

dz3dz5dz6

Vµ1(z1)Vµ2(z2)Vµ3(z3)Vµ4(z4)WNSNS(z5,z6)

Four BoundaryAmplitude

D

trµ(₁αµβ₂)µ¯₃( ˙αµ¯β₄˙)E=_c2πvij

NSI1σ

αα˙

i σ¯

ββ˙

j vij∈(3,3)of SUL(2)×SUR(2)

hσ2(z1)σ2(z2)σ2(z3)σ2(z4)i=f

z14z23

z13z24

z13z24

z12z23z34z41 1/4

f(x) =₍ Λ(x)

F(x)F(1−x))1/2 F(x) =2F1(1/2,1/2;1;x)

Λ(x) =P

n1,n2e −2π

α0

F(1−x)

F(x) n

2 1R

2 1+

F(x)

F(1−x)n

2 2R

2 2

ANS_µ4−NS=h(ER)_[1¯_1]+ (ER)_[3¯_3]

i kikjvij

δ˜g₁¯₁=δ˜g₃¯₃ =−2πi vij k_kikj2

We found again agreement with the SUGRA solution. One can even turn on different condensate to get new SUGRA solutions:

e

Oααβ˙ β˙ _=trµ(α

1 µ¯

( ˙α

2 µ

β)

3 µ¯

˙

β)

4

ˆ

O(αβγ) ˙β _=trµ(α

1 µ

β

2 µ

γ)

3 µ¯

˙

β

Part III

The 4d solution: STU model

L=√g4

R4−P3I=1

∂µUI∂µ_UI¯

2∂ImUI2 −₄1·2!Fµν,ΛF

µν,Λ

The four-dimensional geometries can be viewed as solutions of an N =2 truncation ofN =8 supergravity involving the gravity multiplet and three vector multiplets. The scalarsUIparametrise the

complex structures of the 2-tori.

ds2₄ =−e2U(dt+w)2+e−2U|d~x|2 Aa= (A0,AI) =wa+aa(dt+w)

UI =−bI+i(Ve2UZI)−1

bI = KI

V −

µ

ZI a0=−µV2e4U aI =Ve4U

−Z1Z2Z3

ZI +KIµ

∗₃dw0 =dV ∗3dwI =−dKI

Regularity of the Solutions in 4d

A class of asimptoticallyAdS2×S2×T6geometries (IWP) has been shown to be regular in 4d GR (UI =0).O.Lunin(2015),arxiv.org/abs/1507.06670

Remarkably, this result was achieved for harmonic functions written in terms of an arbitrary profile:

V=LI =Im(H) M=−KI =Re(H) H(~x) =hreg(~x) +R02π 2dvπ_|~x−~_F1_(v)|

r

1+(~x−~F)~A(v) |~x−~F|2

The analog for asymptotically flat solutions turns out to be so easyL.Pieri(2016),1507.06670. Anyway, to evade strong No go theorems

for regular solutions in four dimensional gravity (Gibbons,Warner (2013))

and to make contact with the physics ofAdS2, these black holes should resemble a wormhole like geometry, that is not what we really want... Let’s go to higher dimensions!

The 11d lift (5+6)

The 4d solution lifts to an 11d solution representing intersecting M5-branes with four electric and four magnetic charges.

ds2=ds2₅+ds2_T6 R1,3×S1×T6→ {t, ~x,Ψ,yI,˜yI} ds2₅ =−[dt+µ(dΨ+w0)+w]2

(Z1Z2Z3) 2 3

+ (Z1Z2Z3)

1

3 V−1(dΨ +w₀)2+V|d~x|2

ds_T6 =P3_I=1

Z1Z2Z3

Z3

I 1₃

(dy_I2+d˜y2_I)

Micro-states of the four dimensional black holes can be generically defined as smooth geometries with no horizons or curvature singularities in eleven dimensions carrying the same mass and

Regularity conditions in 5d

Regular solutions can be constructed in terms of multi-center harmonic functions(V,LI,KI,M)with the positions of the centers

and the charges chosen such thatZIare finite andµ=0 near the centers (ri=|~x−~xi|,i=1, ...N).

Multi-center Taub-NUT ansatz V=v0+

N X i=1

qi ri

,LI=`0I+ N X i=1

`I,i ri

,KI =kI₀+

N X i=1

kI_i ri

,M =m0+ N X i=1

mi ri Near each center,R×R4/Z|qi|, asymptoticallyR1,3×S1Ψ.

Geometry factorises, i.e. regular in 5-d, if near the centers ZI

ri≈0 ≈ζ i

I(finite) and µ

ri≈0≈0

Absence of horizons and closed time-like curves requires ZIV >0 and e2U >0

Bubble Equations

ZI finite near the centers if:

`I,i =−

|_IJK|

2 k_iJk_iK

qi

, mi= k

1 i k2i ki3

q2 i

µvanishes near the centers if Bubble Equations are satisfied N

X j=1

Π_ij

rij +v0 k1

i k2i ki3 q2

i

−

3 X I=1

`0IkIi − |IJK| kI

0kJikKi

2qi −m0qi =0 withΠ_ij = (qiqj)−2Q3

I=1

kI_iqj−kI_jqiandrij =|~xi−~xj|

Bubble equations imply absence of Dirac-Misner strings

∗₃dw= 1

2 N X i,j=1

Π_ij 1

rj − 1 rij ! d1 ri = 1 4 N X i,j=1

Π_ijωij

withωij = (~ni+~nij)·(~nj−~nij)dφij/rijfree of DM strings along lines between two centers, since numerator vanishes there.

Asymptotic Charges

M= 1

8πG Z

S2

∞

?4dξ(t) , J =− 1 16πG

Z S2

∞

?4dξ(φ) , Qa = 1

4π

Z S2

∞

(Iab?4Fb− RabFb) , Pa = 1 4π Z S2 ∞ Fa Boundary conditions and charges for orthogonal branes (`I =v=Q/2, M=KI =0 is Reissner-Nordstrom BH):

V≈1+v

r LI≈1+

`_I

r K

I _{≈M ≈o(r}−2₎ M=v+`1+`2+`3,P= (v,0,0,0), Q= (0, `1, `2, `3) Angular Momentum:

H=h0+ h1

r +

~_h

2·~x r3

~_{J =m}

Scaling solutions

If the coefficientsk_iIsatisfy

v0mi− 3 X I=1

`<sub>0I</sub>k<sub>i</sub>I+k<sub>0</sub>I`_Ii−m0qi =0

invariance under rigid rescaling of the positions of the centers

~xi →λ~xi

Multiplying by the positions of the centers~xi, the solution can be shown to carry zero angular momentum in agreement with (Sen’s) expectations for micro-states of single center black holes. In fact this is not obvious (see canonical ensemble

interpretation) and usually~J 6=0 from electric and magnetic charges, even if the black hole is BPS.

Fuzzball of orthogonally intersecting branes BH

Boundary conditions and regularity for single center black hole:

`0I =v0=1 m0=m=k0I =kI=0 N X i=1

kI_i =

N X i=1

k1_i k2_i k_i3=0 Forqi =1 (to avoid orbifold singularities) andNcenters:

P0=N , QI =− N X i=1

|_IJK|kJ_ikK_i 2 Bubble Equations:

N X

j6=i Q3

I=1(kIi −kIj) rij

+k1_ik2_ik3_i−

3 X I=1

kI_i =0

Configurations with one or two centers fail to meet the BPS requirementQI >0. Let us start (and end) with three centers fuzzballs.

Three Center Solutions

The bubble equations for three centers can be solved in general by taking (we needrij >0 and triangle inequalities):

r12=

Π12r23

Π23−r23(Γ2−Λ2) r13=

Π13r23

−Π23+r23(Γ1+ Γ2−Λ1−Λ2)

Πij=

3 Y

I=1

(kI_i−kI_j) Γi=

3 X

I=1

kI_i Λi=k1_ik2_ik3_i

kIi=

_−κ

1κ2 −κ1κ3 κ1(κ2+κ3)

κ3 κ2 −κ2−κ3

−κ4 κ4 0

V=1+

3 X

i=1

1

ri M=κ1κ2κ3κ4 ₁

r1 − 1

r2

L1=1+κ4 _κ

3 r1

−κ2 r2

L2=1+κ1κ4

−κ2 r1

+κ3

r2

L3=1+κ1 _κ

2κ3 r1

+κ2κ3

r2

+(κ2+κ3)

2 r3

K1=κ1

−κ2 r1

−κ3 r2

+κ2+κ3

r3

K2=

κ3 r1

+κ2

r2

−κ2+κ3 r3

K3=κ4 −1 r1 + 1 r2

Scaling Solutions

The scaling solution corresponds to the choice:

κ2=0 κ1=1 κ3=κ4=κ One finds

kIi =   

0 −κ κ

κ 0 −κ

−κ κ 0

 

 r12=r23=r13=`

P0 =3 Q1=Q2=Q3=κ2

for any given`. One can show thatZIV>0 ande−4U >0. Very peculiar solution: centers in an equilateral triangle with

arbitrary size,~J=0, regular and smooth everywhere and stringy interpretationm2+PkI2=0. There are actually 12 solutions, from the allowed permutations of the matrix entries.

Another (non scaling) solution

κ₂=0, κ₁=3κ, κ₃=2κ, κ₄=κ

kIi =   

0 −3κ 3κ

κ 0 −κ

−2κ 2κ 0

  

r12 =

12κ2r23 12κ2_−r

23

r13=

6κ2r23 6κ2_−r

23

P0=3 Q1=2κ2 Q2=6κ2 Q3=3κ2r23<6(2−

√

2)κ2

We notice that triangle inequality in this case impose an upper bound onr23leading to a moduli space of finite volume.

Part IV

The quantum mechanics on the brane world-volume

The 4 intersecting stacks of D-branes enjoy an effective description as aN =4 SQM in 0+1 dimensions.

Untwisted Fields:

Adjoint Chiral SuperMultiplet:

Φ_i(a)={φ_i(a), χ_α,(a)_i,F_i(a)} a=1,2,3,4 i=1,2,3 Adjoint Vector SuperMultiplet:

V(a)={A₀(a),x(_ia), λ_α(a),λ¯(_α˙a),D(a)} i=1,2,3

Twisted Fields:

Bifundamental Chiral SuperMultiplet: Z(ab)={z(ab), µ(_αab),F(ab)}

The Lagrangian

L=

Z

dx0d2θd2θ¯tr h

Z†bae2gVZab+2e−2gVΦa_i†e2gVΦa_i +ξVi+

+

Z

dx0d2θtr ₁

8g2W

α_W

α+W(Φ,Z)

+h.c.

For generic small closed string moduli:

W =

4 X a,b=1

Tr(Z(ab)Φ_i(a)Z(ba)ηi_ab) +

4 X a,b,c=1

Tr(Z(ab)Z(bc)Z(ca))−

−

4 X a,b=1

The functionsc(a)andc(ab)depends on the closed string moduli and characterize the generated Fayet-Iliopoulos and F-terms respectively. At linear order the coupling is computed by the two point string amplitudes on the disk:

A_D,F=DWg,bVD,F E

(1)

W(z1,z2) = (ERa)MNe−ϕcψMeikX(z1)c e−ϕψNeikRX(z2) VD(z3) =ξΩMN:cψMψN(z3) :

VFi(z3) =ζMNi :cψMψN(z3) : leading to

A_D=ξtr(ΩERa)

A_Fi =ζ(i)MN(ERa)[MN]=

1 2ζ

iijk_(ER

Higgs-Coulomb-SUGRA map

In the limitgs→0 we first go from a SUGRA multiple centers configuration in the large scale regime to a Quiver QM

description. Indeed after quantum corrections, the Coulomb branch (x(_ia) 6=0) of the QQM is actually identical to the SUGRA “solution space”, as both are subject to the Bubble equations.

If we keep on loweringgs, the system decay into a configuration with nonzero vevs forz(ab): Coulomb→Higgs. Finally, at gs=0, thez(ab)can be interpreted as the moduli of a suitable geometric object, a system of intersecting susy D-BranesDenef (2002).

pure-Higgs microstates

States in the Higgs branch can be represented as cohomology classes in the moduli space of classical solution ( D and F constraints).

.

The Higgs-Coulomb/SUGRA map identifiesSU(2)action on the cohomology with space-time angular momentum.

For more than 2 centers, the Higgs-Coulomb map is not 1:1 anymore, in fact there are an exponential number of middle cohomology (zero angular momentum) pure-Higgs states.

Bena, Berkooz, de Boer, El-Showk , Van den Bleeken (2012)

.

Pure-Higgs states can have a SUGRA description as scaling solutions.

What has been done

Explicit counting of the number of ground states for low number of branes in each stack.(Chowdhury, Garavuso, Mondal, Sen (2015,2016))

VF=

4 X

a=1 3 X i=1 ∂W

∂Φ(_ia)

2 + 4 X

a=1 4 X

b=1

∂W

∂Z(ab)

2

VD=1 2

4 X

a=1 4 X

b=1

(¯Z(ab)Z(ab)−¯_Z(ba)

Z(ba))−c(a) !2

VGauge=

3 X

i=1 4 X

a=1 4 X

b=1

(x(_ia)−x_i(b))(x(_ia)−x_i(b))(¯Z(ab)Z(ab)+ ¯Z(ba)Z(ba))

BPS ground states = solutions toVF+VD+VG=0. For generic closed moduli one finds a zero dimensional space of solutions, therefore only middle cohomology. Using the BPS index of D1-D5-KK-P one can confirm that~J =0.

This analysis is completely classical! Are they different microstates in gravity?

What one could do (Toy SQM example)

L= 1

2x˙ 2₋1

2(∂xW(x)) 2₊ i

2( ¯ψ

˙

ψ−ψψ˙¯ )−∂_x2W(x) ¯ψψ

δx=ψ¯−ψ¯ δψ=(ix˙+∂xW) δψ¯= ¯(−ix˙+∂xW)

δ

Z Ldt=

Z

dt−i˙Q−i˙¯Q¯

Q= ¯ψ(ix˙+∂xW) = ¯ψ(ip+∂xW) Q¯ =ψ(−ix˙+∂xW) =ψ(−ip+∂xW)

[x,p] =i {ψ,ψ}¯ =1 ψ|0i=0 Ψ =f1(x)|0i+f2(x) ¯ψ|0i

Quantum Ground State

W(x) =W(xi) +1 2

h _∂2_W

∂xA_xB i

x=_xi

(xA−xA0)(x B₋

xB0) +· · ·=W(xi) + X

A

c(_A0)(τ₍A₀₎)2

Ψ_i=e− P

A |c

(i)

A |(τ(Ai))

2 _Y

B:c(_Bi)<0

¯

ψB|0i

For every BPS ground states, one can find the associated wavefunction. Then one can compute the condensates appearing in

the stringy computation, relating QM with SUGRA.

Qα=

X

a6=b

2z˙¯(ba)µ(_αab)+iσ_αi_β˙(¯σ0)ββ˙ xi(ab)¯z

(ba)_µ(ab) β −iµ¯

(ba) ˙ α (¯σ

0₎αβ˙ αβ

∂W ∂¯z(ba)

Open String Condensate

We promote the polarizations of vertex operators to QM operators.

Vµα =µ(αab)e− ϕ 2 Sαe

i

2hjab Y

j6=jab

∆_j {µ¯(_α˙ba), µ(_βab)}=δβα˙

In the string amplitude we should compute both the CFT correlator and the QM one.

hVµ1(z1)Vµ2(z2)Vµ3(z3)Vµ4(z4)WNSNS(z5,z6)i=

h. . .i_CFT hΨ_i|µ(a1a2)

A1 µ

(a2a3)

A2 µ

(a4a1)

A3 µ

(a4a1)

A4 |Ψii

hΨ_i|µ(a1a2)

A1 µ

(a2a3)

A2 µ

(a4a1)

A3 µ

(a4a1)

Summary and Conclusions

Black hole microstates can be identified in gravity by computing the backreaction of the branes with disk scattering amplitudes computations.

Large class of regular and horizonless solutions in supergravity have been found. In this work we have

focused on four charge geometries. Many puzzles remains, in particular one can ask how much general are these results and if SUGRA is enough for fuzzballs.

The condensate of open strings binding the branes are actually not arbitrary numbers, but they are fixed by the explicit quantum configuration of the microstate. In a SQM approach one can hope to uncover the difference between microstates. Will the microstates differ for higher order multipoles in the gravity description?

Future

Three Charge Black Holes: More "down to earth" calculations : infalling observer, fuzzball formation, scattering of probes

.

Four Charge Black Holes: More microstates (from SQM, MSW CFT, scattering amplitudes and SUGRA solutions). Would be nice to useAdS2/CFT1.

.